New Simple Methods of Tuning Three-Term Controllers for Dead-Time Processes

Transcription

1 New Siple Methods of Tuig Three-Ter Cotrollers for Dead-Tie Processes.G.ARVANTS Departet of Natural Resources Maageet ad Agricultural Egieerig Agricultural Uiversity of Athes era Odos 75, Botaikos 8 55, Athes GREECE G.D.PASGANOS, G.ALOGEROPOULOS Departet of Matheatics Uiversity of Athes Paepistiiopolis 5784, Athes GREECE Abstract: - New siple ethods are preseted for tuig three-ter cotrollers for dead-tie processes. The ethods are based o appropriate aipulatios of the expoetial dead-tie ter i the deoiator of the closed-loop trasfer fuctio for a servo proble, ad, subsequetly, o appropriately atchig the coefficiets of correspodig powers of s i the uerators of the resultig trasfer fuctios ad those i their deoiators. This techique gives siple algebraic systes of equatios for the cotroller paraeters. The proposed ethods are uiversal ad ca readily be applied to a wide variety of liear process odels with tie delay. Siulatio results show that our ethods provide iproved perforaces as copared to the ost recet tuig ethods. Moreover, the cotroller settigs obtaied by the proposed ethods give a robust perforace for ucertaity i the process odel paraeters. ey-words: - Pseudo-derivative feedback, o-lie cotroller tuig, process cotrol, dead-tie processes. troductio Tie delays (also kow as trasport lags, dead ties or tie lag ay arise i physical, cheical, biological, electroic ad ecooic processes ad systes, as well as i the process of easureet ad coputatio [. [, it has bee poited out that apparet tie delays result whe actual trasportatio or easureet delays are preset, or whe high order processes are approxiated by eas of lower order trasfer fuctios. Note also that, a tie delay ay appear to be preset due to the effect of the cobiatio of a large uber of tie lags. Whatever the case is, tie delays are preset i ost realistic processes ad systes ad ust be carefully take ito accout whe odel the. Process odelig iheretly ivolves a coproise betwee odel accuracy ad coplexity o oe had, ad the cost ad effort required to develop the odel, o the other had. the process cotrol circles, there is uch debate over how coplex a odel ay reasoably be. The questio as to whether the process is best odelled by a o-liear odel with tie delay or a liear odel with tie delay is also iportat, as it affects the coplexity of the cotrol proble. For the purpose of desigig cotrollers, it appears that it is reasoable to suggest that, eve if the process has o physical tie delay, it ay be possible to odel such a (possibly high order) process by a liear low order odel plus tie delay. t appears also reasoable that either a first order plus dead tie (FOPDT) odel (for stable overdaped processe or a secod order plus dead tie (SOPDT) odel (for stable overdaped or uderdaped processe, or fially a itegrator plus dead tie odel (i case of processes with very large tie costat should be cosidered [-[5. Either of these approxiate process odels would appear to be sufficietly accurate for ay applicatios. Of course, there are exceptios fro this typical situatio. deed, although ost processes exhibit ope loop stable behavior, several others exhibit

2 ultiple steady states due to syste oliearities. Soe of these steady states ay be ustable. For several reasos, like safety, axiizatio of productivity ad reductio of ecooic costs, it is ofte desirable to operate such processes aroud their ustable steady states. To approxiate the ope loop dyaics of such systes, ay of these processes ca be satisfactorily described by ustable trasfer fuctio odels, the ost popular beig the ustable first order plus dead-tie (UFOPDT) odel ad the ustable secod order plus dead tie (USOPDT) odel (with either oe or two ustable pole [6. O the other had, wheever a process cotrolled variable ecouters two (or ore) copetig dyaic effects, with differet tie costats, fro the sae aipulated variable, the resultig coposite dyaic behaviour of this process ca exhibit a troublesoe iverse respose or a large overshoot respose (which correspods to a positive or egative zero, respectively, i the ope-loop trasfer fuctio) [7, [8. Cotrol ethods for dead-tie processes ay be broadly classified ito two ai categories: The first category cosists of ethods based o structurally optiized cotrollers. The secod category icludes ethods based o paraeter optiized cotrollers. the first class, the cotroller structure ad paraeters are adapted optially to the structure ad paraeters of the process odel. the secod class, the cotroller paraeters are adapted to the cotroller structure. The three-ter Proportioal-tegral-Derivative (PD) cotroller is the ost coo type of paraeter optiized cotroller. the process cotrol, ore tha 95% of the cotrol loops are of the PD type [5. The ai reaso is its relatively siple structure, which ca be easily uderstood ad ipleeted i practice, ad its ability to copesate ay practical idustrial processes. Several variatios of the so-called ideal or stadard PD cotroller have bee proposed (for a review see [9). Moreover, over the years, a variety of tuig ethods for desigig the paraeters of PD cotrollers have bee reported i the literature. The iterested reader ay refer to [9 for a detailed review of these ethods. The available tuig ethods ay be broadly classified ito six ai categories: (a) Process reactio curve ethods, (b) ethods based o iiizig a appropriate perforace criterio, (c) direct sythesis tuig ethods, (d) ultiate cycle tuig ethods, (e) robust tuig ethods with a explicit robust stability ad robust perforace criterio, ad (f) tuig ethods based o several specific desig criteria, like the quarter decay ratio specificatio, or the agitude ad frequecy iforatio at a particular phase lag. ay of these ethods, oe or two adjustable paraeters are used to calculate the PD settigs. ay cases, the desig of PD cotrollers for delayed processes is based o ethods that were origially used for the cotroller desig of delayfree processes. Moreover, ost of the PD tuig ethods reported i the literature are suitable oly for stable dead-tie processes, although i recet years soe specific ethods are proposed for tuig three-ter cotrollers for itegratig ad ustable dead-tie processes, as well as for iverse respose or large overshoot processes with dead tie. Few of these tuig ethods are o-odel specific, i.e. they are applicable regardless of the odel of the process uder cotrol. Fially, i the ajority of the available tuig ethods, the desig procedure is soewhat coplicated. Therefore, there is a eed for siple uiversal tuig ethods with iproved perforaces. The focus of this paper is the cotrol of deadtie processes usig three-ter cotrollers. The particular cotroller structure cosidered i the paper for the cotrol of tie delayed processes is the Pseudo-Derivative Feedback (PDF) cotroller structure, origially proposed i [. As it is explaied i Sectio, the PDF cotroller is a variatio of the stadard PD cotroller. The reaso for usig PDF cotrol istead of PD is aily due to its advatages over covetioal PD cotrol. The ai of the are: (a) The PDF cotroller ca provide the closed-loop trasfer fuctio of the syste with the absece of uerator dyaics. Therefore, it aturally raps the cotroller effort, sice it iteralizes the set-poit filter that oe would apply to cacel the udesired zeros, itroduced i the PD cotroller cofiguratio, which produce excessive overshoot. (b) the PDF cotroller, the D-actio is oved fro the direct cotrol brach ito the feedback, thus keepig the advatages of the differetial actio, but without difficulties caused by a differetiator placed i the direct brach (e.g., the differetial peak i the cotrol variable or the icorrect calculatio of the derivative due to physical liitatios of the differetiatig device, etc.). (c) The PDF cotroller, with the itegral actio i the direct brach, is coveiet for the reaso of obeyig the "oe aster priciple" [, i.e. the priciple of oe actio i the direct brach. This prevets the closedloop syste fro several deficiecies [. this work, ew siple ethods are preseted for tuig PDF cotrollers for dead-tie processes. The ethods are based o appropriate aipulatios of the expoetial dead-tie ter i the deo-

3 iator of the closed-loop trasfer fuctio for a servo proble, ad, subsequetly, o atchig the coefficiet of correspodig powers of s i the uerators of the resultig trasfer fuctios ad that i their deoiators. This techique gives siple algebraic systes of equatios, the solutio of which provides the PDF cotroller settigs i ters of the paraeters of the process odel ad o two adjustable paraeters. The proposed ethods are uiversal ad ca readily be applied to a wide variety of liear process odels with tie delay, fro FOPDT odels to ustable SOPDT odels with a ustable zero. Although it is practically ipossible to copare the proposed ethods with the hudreds of ethods reported i the literature [9, siulatio results show that our ethods provide iproved perforaces as copared to the ost recet tuig ethods. Moreover, the cotroller settigs obtaied by the proposed ethods give a robust perforace for ucertaity i the process odel paraeters. Three-Ter Cotroller Structures for Dead-Tie Processes the literature, several cotrol techiques have bee proposed for the cotrol of dead-tie systes. The ost widely used are alterative schees of the so-called PD or three-ter cotroller. The cotiuous tie ideal or stadard PD cotroller for a sigle-iput, sigle-output process odel has the followig trasfer fuctio. G C = C τ Ds () τ s where C is the proportioal gai, τ is the itegral tie costat ad τd is the derivative tie costat. t is, however, ucoo to ipleet the PD cotroller structure provided above i practice. A uber of alterative odified PD cotroller structures are used i the process cotrol literature, ad of course i idustrial practice. The ai of the are [, [9: (i) The ideal PD cotroller with first or secod order filter, (ii) several types of the ideal PD cotroller with set-poit weightig with or without first order filter, (iii) the so-called classical PD cotroller, (iv) several types of the o-iteractig PD cotroller, (v) the idustrial PD cotroller, (vi) the series PD cotroller with or without filtered derivative, (vii) the series PD cotroller with or without lead eleet, (viii) the PD cotroller with filtered derivative, (ix) the stadard or SA for PD cotroller, (x) several types of the two degree of freedo PD cotroller, ad (xi) the -PD cotroller (also called the Pseudo- PDF-Cotroller L( R( E( U( s s d P G P( Y( Figure. The Pseudo-Derivative Feedback cotrol cofiguratio. PD-Cotroller Pre-Filter L( R( E( U( C sτ G SPF( _ D G τ P( Ιs Y( Figure. deal PD cotroller with set-poit filter equivalet to the PDF cotrol structure. Derivative Feedback cotroller). Each particular variatio of the PD cotroller has its ow advatages ad disadvatages, which will ot be reviewed further here, because a copariso of the differet PD cotroller types is beyod the scope of the preset paper. However, sice the focus of this paper is o the derivatio of ew ethods of tuig a particular class of three-ter cotrollers, we will ext briefly review this cotrol structure, kow as the Pseudo- Derivative Feedback (PDF) cotrol. The PDF cotroller has origially bee proposed i [, ad it is illustrated scheatically i Figure. t is ofte called the -PD cotroller, due to the fact that the three cotroller actios are separated. tegral actio (which is dedicated to steady state error eliiatio) is located i the forward path of the loop, whereas proportioal ad derivative actios (which are aily dedicated i assigig the desired closed-loop perforace i ters of stability, resposiveess, disturbace atteuatio, etc)) are located i the feedback path. This separatio leads to a better uderstadig of the role of each particular cotroller actio. Moreover, the PDF cotroller has soe distict advatages over the ideal PD cotroller of the for (). The ai of the are: (a) A first advatage stes fro the fact that PDF cotrollers ca provide the closed-loop trasfer fuctio of the syste with the absece of uerator dyaics. So, the speed respose characterized by dapig ratio ad atural frequecy ca be easily predicted with adjusted cotrol gais. other aalytic view, the PDF cotroller results i a equivalet trasfer fuctio to the PD cotroller of the for (), havig filtered iput coad, ad the poles of the secod-order set-poit filter are cacelled out with the zeros itroduced by the PD cotroller. Therefore, the set-poit filter is of the for

4 GSPF = () τ Dτ s τ s The PDF cotroller aturally raps the cotroller effort, sice it iteralizes the pre-filter that oe would apply to cacel the udesired zeros itroduced i the PD cotroller cofiguratio, which produce excessive overshoot. The equivalece of the PDF ad the ideal PD cotroller with set-poit filter is show i Figure ad it is based o the followig relatios betwee the paraeters of the two alteratives cotrol schees P = C, = C / τ, d = C τ D () t is worth oticig at this poit that, i the case of regulatory cotrol the two alterative cotrol schees preseted above are idetical i ters of perforace, provided that relatios () hold. (b) Aother fudaetal advatage of the PDF cotroller over PD cotrol lies i ovig the D- actio fro the direct cotrol brach ito the feedback, thus keepig the advatages of the differetial actio, but without difficulties caused by a differetiator placed i the direct brach. Naely, it is kow that the D-actio ca be used for icreasig the speed of the syste s respose. The reaso for ovig it for the direct brach is that it causes the sudde chage i the error sigal, where the physiccal liitatios of the differetiatig device cause the icorrect calculatio of the derivative, thus, accordigly, the real perforace of the syste will be saller tha the ideal perforace set by the odel. Also, the presece of the D-actio ad to the lesser extet also the P-actio, i stepwise variatio of the referece (iput) sigal, ca cause the so called differetial peak i the cotrol variable, which caot be hadled physically by the ajority of the executive orgas. the PDF cotrol the D-actio is oved ito the feedback by the output, ad sice the output value represets the result of several itegratios, it will vary slower tha the other sigals i the syste, ad thus the differetiator s respose will be ore realistic. (c) Fially, the applicatio of the PDF cotrol is also coveiet fro the reaso of obeyig the "oe aster priciple" (accordig to teriology used i [), aely the priciple of oe actio i the direct brach. The deficiecy of the PD cotrol laws faily is also i that, sice the cotroller is required to siultaeously respod o sigals that ca be coflictig. f for exaple the error sigal (which is hadled by the PD cotroller to produce the cotrol iput) is the siusoidal sigal, the its derivative ad itegral are oved for 9 o ad 9 o with respect to the error sigal, respectively, so the cotroller is forced to siultaeously process three differet sigals ad geerate the cotrol iput. Result of such a aalysis, accordig to [, is the coclusio that the ost coveiet is for the cotrol algorith ot to cotai ore tha oe actio i the direct brach. The applicatio of this rule actually represets obeyig "oe aster priciple". Cosiderig the previously preseted aalysis of the D-actio, it is obvious that its applicatio i the direct brach should be avoided. O the other had, the P-actio of the PDF cotroller ca be placed i the direct brach, oly i the case where there is o disturbig actio, ad where the fast respose is ot required. the case of existece of disturbace ad oly of the P-actio i the direct brach, the syste would always preset a certai error i the statioary state. Aother deficiecy of P-actio applicatio is also i that it is ot realistic to cosider that the cotrol eleet istataeously respods to the stepwise respose. Thus, as the ost coveiet solutio will be the placeet of the itegral actio i the direct brach. Due to its advatages over PD cotrol, we shall ext focus our attetio to the PDF cotrol structure ad propose ew siple ethods for tuig its settigs, i cases where it is applied to cotrol a variety of liear tie delayed process odels. The proposed tuig ethods Cosider the geeral trasfer fuctio odel of the for q( G P = exp( d (4) p( of a dead-tie process, where p R [ s ad q R [ s are polyoials of the ideteriate s, with. Let us ow apply to syste (4) the PDF cotrol structure depicted i Figure. t is ot difficult to see that the closed-loop trasfer fuctio fro referece R( to output Y( is give by q( exp( d GCL = (5) sp( q( ( d s P s ) exp( d We ext propose four ew ethods for tuig the PDF cotroller settigs. These ethods are as follows:. Method Re-writig the expoetial ter i the deoiator of (5) i the for

5 [( a ds exp( ds ) = exp( ad / exp ) for soe a R, relatio (5) takes o the for qcl,( GCL = exp( ad (6) pcl, where q q( ) p cl, = q( ( cl, = s sp( exp[ ( a) ds ( s s ) exp( a d d P Let us ow cosider the uerator ad the deoiator ters of equatio (6) usig the Taylor series expasio for q cl, ( ad p cl, (. The coefficiet of the costat ter (coefficiet of s ) of the uerator is already equal to that of the deoiator because of the presece of the itegral actio. Sice the objective of the cotroller is to ake Y(/R(=, i order to obtai the cotroller settigs, oe should equate the coefficiets of like powers of s of the uerator ad the deoiator of (6). However, sice the desig specificatios for stable systes caot usually be et by ustable systes or by systes with stable ad ustable opeloop zero dyaics, oe caot geerally force Y(/R( as havig the value. For this reaso, i the sequel, we shall equate the coefficiet of the correspodig powers of s of the uerator with that of the deoiator ultiplied by a factor b R, except for the coefficiet of s. Thus, o equatig the coefficiets of s, we get () q () () { p() [ q () a dq() q() } = b i P or, equivaletly d T p = b p() (7a) where = b q() (7b) () [ ( b ) q () badq() Siilarly, o equatig the coefficiets of s, after soe straightforward aipulatios, we get d T () p = b [ p () ( a ) dp() (8a) b q() () = b [ q () adq() (8b), () () {( b ) q () b [ a dq () a d ()}, = q (8c) Fially, o equatig the coefficiets of s, we get d T () () p = b [ p () ( a ) dp () (9a) ( a ) d p() where () 6b [ q () adq() =, (9b), () () = b q () a dq () a d () (9c) [ () = { ( b ) q () () () [ a dq () a d q () a d q() }, q, (9d) b Relatios (7a), (8a) ad (9a) costitute a set of liear algebraic equatios with respect to d, p ad,, havig the for Nk = z () where T d T N =, k = p T b p() () z = [ b p () ( a) dp() () () b [ p () ( a ) dp () ( a ) d p() The solutio of () provides the PDF cotroller settigs sought. t is worth oticig at this poit that, i the case where the process odel has o zeros, the q (i) ()=, for i=,,. The, as it ca be easily checked by relatios (7)-(9), paraeter b ca be cacelled-out i both sides of (), ad does ot play ay role. this case, oly the adjustable paraeter a is used i the ethod. However, as it will be show later o i sectio 4, the adjustable paraeter b plays a iportat role i cases where the process odel cotais at least oe zero. Sice, i this subsectio, we preset a geeral for of Method applicable to systes with or without zeros, we ext use a typical value b = for this adjustable paraeter i the case of systes without zeros, although ay other value of it is also acceptable.. proved Method this ethod, we still equate the coefficiet of power of s i the uerator of (6) to b ties that of the deoiator whereas the coefficiets of s ad s is set to c (=βb ) ties that of the deoiator. Now, there are three tuig paraeters: a, b ad β.

6 However, to ake the tuig procedure siple we ca keep β to a stadard value (say β=). Hece, there are oce agai oly two tuig paraeters a ad b. Siilar to the steps give i the previous subsectio for Method, we get the followig liear algebraic equatios for the PDF settigs as: N k = z where T d T N =, k = p T cq() () = c[ q () adq(), () () = ( c) q () c a dq () a d () { [ } () 6c[ q () a dq(), q = = c q,, () () [ () adq () a d () () = { ( c) q () () () [ a dq () a d q () a d q() }, q, c b p() () z = [ c p () ( a ) dp() () () c[ p () ( a ) dp () ( a ) d p() Note that, i the case of systes without zeros, the iproved Method gives exactly the sae cotroller settigs as those obtaied by Method. This is due to the fact that i this case, both b ad c ca be cacelled out i both sides of the cotroller desig equatio, ad hece, oce agai, it reais oly oe adjustable paraeter: a. However, as it will be verified i the ext sectio, i the case of systes with zeros, the proposed iproved ethod gives a ehaced perforace as copared to Method, sice the, both b ad c are preset ad play a very iportat role.. Method Re-writig the expoetial ter i the deoiator of (5) i the for exp( ds ) = exp( ad / exp[ ( a ) ds for soe a R, relatio (5) takes o the for qcl, GCL = exp( d () pcl, where q = q( exp ( a ds [ cl, ( ) p cl, q( = sp( exp[ ( a ) ds ( s s ) exp( a d d P Usig a arguet aalogous to that of Method, we ext equate the coefficiets of the correspoddig powers of s of the uerator of () with that of its deoiator ultiplied by a factor b R (except for the coefficiet of s ). Thus, o equatig the coefficiets of s, we get d T p = b p() (a) with = b q() (b) () { ( b ) q () [ d ( b ) ad q() } Siilarly, o equatig the coefficiets of s, after soe straightforward aipulatios, we get d T () p = b [ p () ( a ) dp() (a) where b q() () = b [ q () adq() (b), (), = { ( b ) q () () [( a ) ba dq [( a ) b a d q() } () (c) Fially, o equatig the coefficiets of s, we get d T () () p = b [ p () ( a ) dp () ( a ) d p() (4a) where () 6b [ q () adq() =, (4b), () () = b q () a dq () a d () (4c) [ () = { ( b ) q (), q, [ ( a ) b a () [ ( a) b a d q [( a ) b a d q() } dq () () () (4d) Relatios (a), (a) ad (4a) costitute a set of liear algebraic equatios with respect to d, p

7 ad,, havig the for Mk = h (5) where T d T M =, k = p T b p() () h = b [ p () ( a ) dp() () () b [ p () ( a ) dp () ( a ) d p() The solutio of (5) provides the PDF cotroller settigs sought. Note that, as it ca be easily checked by a siple ispectio of relatios ()- (4), i the case of Method, the adjustable paraeter b is ivolved i the cotroller desig equatio (5), eve i cases where the process odel has o zeros..4 proved Method this ethod, we still equate the coefficiet of power of s i the uerator of () to b ties that of the deoiator whereas the coefficiets of s ad s is set to b (=γb ) ties that of the deoiator. Now, there are three tuig paraeters: a, b ad γ. However, to ake the tuig procedure siple we ca keep γ to a stadard value (say γ=5.55). Hece, there are oce agai oly two tuig paraeters a ad b. Siilar to the steps give i the previous subsectio for Method, we get the followig liear algebraic equatios for the PDF settigs as: M k = h where T d T M =, k = p T with b q() () = b [ q () adq(),, = () {( b ) q () [( a ) b a [( a ) ba d q() } () 6b [ q () a dq(), = = b q,, dq () () [ () a dq () a d (), q = () {( b ) q () [ ( a ) b a () () [ ( a ) b a d q () [( a ) b a d q() } dq () () () b p() () h = b [ p () ( a ) dp() () () b [ p () ( a ) dp () ( a ) d p() As it will be verified i the ext sectio, the proposed ethod gives a iproved perforace as copared to Method. 4 Siulatio Results We ext apply the tuig ethods preseted i the previous sectio to a variety of dead-tie processes, i order to illustrate its wide applicability i facig diverse process characteristics. 4. Exaple Let us first cosider a stable first order plus dead tie (FOPDT) odel of the for G P = exp( d /( Ts ) with =, T= ad d=.5, which is a typical exaple for testig tuig ethods for three-ter cotrollers [. The proposed Method is applied here by selectig a =. ad b =. With this selectio, the PDF cotroller settigs obtaied by Method are P =.99, =.874, d =.45. The PDF cotroller settigs obtaied whe applyig Method with a =. ad b =5 are P =.785, =.9986, d =.8. The settigs of a PD cotroller tued accordig to the ethod reported i [ are C =.5, τ i =.5 ad τ D =.67. Note that the ethod proposed i [ has favorably bee copared to other well kow tuig ethods for FOPDT odels (e.g. the ope-loop Ziegler-Nichols ad the MC tuig ethod. Figure shows the copariso of the servo-respose of the proposed ethods with that of the ethod reported i [, as well as the perforace copariso of the above Figure. Copariso for stable FOPDT systes. Solid blue lie: Method, solid black lie: Method, dotted lie: Method of [.

8 Table. SE, AE ad TAE values for Exaple. Criterio Method Method Method of [ SE-servo AE-servo TAEservo SE-load AE-load TAE-load Figure 4. Closed-loop resposes uder paraetric ucertaity i. Other leged as i Fig.. ethods for the regulatory cotrol proble. The servo perforace obtaied by our ethods is better tha that obtaied by the ethod i [, i ters of overshoot ad settlig tie, while all three ethods are siilar i ters of regulatory cotrol perforace. Moreover, Table gives the SE, AE ad TAE values for all three ethods. Obviously, the SE, AE ad TAE values for the ethod i [ are saller tha that obtaied by the preset ethods i the case of the servo-proble. The sae holds for the SE value i the case of the regulatory proble, whereas, i the regulatory proble case, the values of the AE ad TAE obtaied by our ethods are saller tha those obtaied by the ethod i [. Fially, the robustess of the proposed ethods is evaluated by allowig a % ucertaity i the process gai. Figure 4 shows the resposes i this case. Siilar resposes are obtaied i the case of a % ucertaity i T or d. The robust perforaces obtaied by the proposed ethods are better tha that obtaied by the ethod i [. particular, it is easy for the iterested reader to check that our ethods ca tolerate ore tha % decreasig ucertaity i T, whereas the ethod i [ does ot. 4. Exaple Let us ow cosider a stable secod order plus dead tie (SOPDT) odel of the for G P = exp( d /( T s )( Ts ) with =, d=.5, T =, T =. The proposed Method is applied here by selectig a =4.5 ad b =. With this selectio, the PDF cotroller settigs obtaied by Method are P =4.485, =.46, d = The PDF cotroller settigs obtaied whe applyig Method with a =4.5 ad b =5 are P =4.456, =.68, d =.686. The settigs of a PD cotroller tued accordig to the ethod reported i [4 are C =.5, τ i =.5 ad τ D =.5, with the tuig paraeter i [4 havig the value λ=.5, i order to obtai a desired closed-loop respose of the for exp(-d/(λ. Figure 5 shows the copariso of the servo-respose of the proposed ethods with that of the ethod reported i [4, as well as the perforace copariso of the above ethods for the regulatory cotrol proble. The perforace obtaied by our ethods, i the case of the servo proble, is better tha that obtaied by the ethod i [4, i ters of overshoot ad settlig tie. Our ethods provide a cosiderably better regulatory cotrol perforace. Table gives the SE, AE ad TAE values for all three ethods. Obviously, the SE, AE ad TAE values for our ethods are saller tha those obtaied by the ethod i [4, for both the servoproble ad the regulatory proble, except for the case of the SE-servo criterio. Note that i [4, a alterative tuig ethod is reported, which provides a closed-loop perforace of the for exp(-d/(λ. The PD settig obtaied by this alterative ethod are C =.9444, τ i =.967 ad τ D =.64, with the tuig paraeter λ=.5. Figure Figure 5. Copariso for stable SOPDT systes. Solid blue lie: Method, solid black lie: Method, dotted lie: Method of [4.

9 Table. SE, AE ad TAE values for Exaple. Criterio Method Method Method of [4 SE-servo AE-servo TAEservo SE-load AE-load TAE-load Figure 6. Copariso for stable SOPDT systes. Solid blue lie: Method, solid black lie: Method, dotted lie: Alterative ethod of [4. 6 gives the copariso of the obtaied resposes. Obviously, our ethods are favorably copared to the ethod give i [4, especially i the case of the regulatory proble. 4. Exaple Let us ext cosider a itegratig plus dead tie (PDT) odel of the for GP = P exp( d / s with =, d= (a typical exaple for testig tuig ethods for such process odel. The proposed Method is applied here by selectig a =. ad b =. With this selectio, the PDF cotroller settigs obtaied by Method are P =.4, =.49, d =.4. The PDF cotroller settigs obtaied whe applyig Method with a =. ad b =5 are P =.66, =.55, d =.4. The settigs of a PD cotroller tued accordig to the ethod reported i [5 are C =.46, τ i =4.5 ad τ D =.45. Figure 7 shows the copariso of the servo-respose of the proposed ethods with that of the ethod reported i [5, as well as the perforace copariso of the above ethods for the regulatory cotrol proble. The servo perforace obtaied by our ethods is better tha that obtaied Figure 7. Copariso for PDT systes. Solid blue lie: Method, solid black lie: Method, dotted lie: Method of [5. Table. SE, AE ad TAE values for Exaple. Criterio Method Method Method of [5 SE-servo AE-servo TAEservo SE-load AE-load TAE-load Figure 8. Closed-loop resposes for PDT odels uder ucertaity i. Other leged as i Fig. 6. by the ethod i [5, i ters of overshoot ad settlig tie, while all three ethods are siilar i ters of regulatory cotrol perforace. Table gives the SE, AE ad TAE values obtaied by the three ethods i copariso, for both the servo ad the regulatory cotrol proble. The AE ad TAE values obtaied by our ethods are saller tha those obtaied by the ethod i [5, which is better i the case of SE. Fially, the robustess of the

10 proposed ethods is evaluated by allowig a 5% ucertaity i the process gai. Figure 8 shows the resposes i this case. The robust perforaces obtaied by the proposed ethods are better tha that obtaied by the ethod i [5. However, for a 5% ucertaity i d the proposed ethods see to be less robust tha the ethod reported i [ Exaple 4 Let us ext cosider a double itegratig plus dead tie ( PDT) odel of the for GP = P exp( d / s Tuig ethods of PD cotrollers for such processes are very liited. [6, a ethod is proposed i order to tue a cotroller of the for GC = c ( / τ i E( c ( b ) R( cτ d sy but the cotroller ad the desig ethod is quite coplicated. Several ethods i order to tue a ideal PD cotroller for PDT processes have bee preseted oly i [7. We ext cosider the exaple treated i [7 i order to evaluate the proposed ethods. this exaple, =.574 ad d=.57. The proposed Method is applied here by selectig a =4.5 ad b =. With this selectio, the PDF cotroller settigs obtaied by Method are P =.499, =., d =.567. The PDF cotroller settigs obtaied whe applyig Method with a =4.5 ad b =5 are P =.569, =.68, d =.565. The settigs of a PD cotroller tued accordig to the ethod reported i [7 are C =.5588, τ i = ad τ D = Figure 9 shows the copariso of the servo-respose of the proposed ethods with that of the ethod reported i [7, as well as the perforace copariso of the above ethods for the regulatory cotrol proble. All Figure 9. Copariso for PDT systes. Solid blue lie: Method, solid black lie: Method, dotted lie: Method of [7. Table 4. SE, AE ad TAE values for Exaple 4. Criterio Method Method Method of [7 SE-servo AE-servo TAEservo SE-load AE-load TAE-load three ethods are siilar i ters of overshoot, while our ethods are better i ters of settlig tie. However, i case of regulatory cotrol a saller error is obtaied whe the ethod i [7 is applied. Table 4 gives the SE, AE ad TAE values obtaied by the three ethods i copariso, for both the servo ad the regulatory cotrol proble. geeral, cosiderably saller values of the above criteria are obtaied whe applyig the proposed ethods. Fially, ote that, as it ca be easily checked by the iterested reader, our ethods ca tolerate a % ucertaity i the process gai or i the process delay, whereas the ethod i [7 does ot, givig a ustable respose i both cases. 4.5 Exaple 5 Let us ext cosider a first order lag plus itegral plus dead tie (FOLPDT) odel of the for G P = exp( d /[ s( Ts ) with =, T=, d=. The proposed Method is applied here by selectig a =4 ad b =. With this selectio, the PDF cotroller settigs obtaied by Method are P =.8, =.5, d =.5. The PDF cotroller settigs obtaied whe applyig Method with a =4 ad b =5 are P =.86, =.76, d =.88. The settigs of a PD cotroller with derivative filterig of the for Gc = c ( (/ τ i ) s τ D ( / ( τ f s ) tued accordig to the ethod reported i [8 are C =.449, τ i =, τ D =.967, τ f =.55. Figure shows the copariso of the servo-resposes of the proposed ethods with that of the ethod reported i [8, as well as the perforace copariso of the above ethods for the regulatory cotrol proble. The servo perforace obtaied by our ethods is better tha that obtaied by the ethod i [8, i ters of overshoot. Our ethods also give a saller axiu error i case of regulatory cotrol. Table 5 gives the SE, AE ad TAE values obtaied by the three ethods i copariso, for both the servo ad the regulatory cotrol proble. The values obtaied by our ethods are

11 Figure. Copariso for FOLPDT systes. Solid blue lie: Method, solid black lie: Method, dotted lie: Method of [8. Table 5. SE, AE ad TAE values for Exaple 5. Criterio Method Method Method of [8 SE-servo AE-servo TAEservo SE-load AE-load TAE-load saller tha those obtaied by the ethod reported i [8 (except for the SE-servo case). Fially, all ethods are siilar i ters of robustess. As it ca be easily checked by the iterested reader, all three ethods ca readily tolerate a siultaeous % ucertaity i all process paraeters. 4.6 Exaple 6 Cosider ext a ustable first order plus dead tie (UFOPDT) odel of the for G P = exp( d /( Ts ) with =, T= ad d=.5. This is a typical exaple for testig tuig ethods for such process odels. Method is applied here by selectig a = ad b =. With this selectio, the PDF cotroller settigs obtaied by Method are P =, =.6667, d =.5. The PDF cotroller settigs obtaied whe applyig Method with a = ad b =5 are P =., =.74, d =.54. The settigs of a PD cotroller tued accordig to the ethod reported i [ are C =.945, τ i =.779 ad τ D =.56. Figure shows the copariso of the servo-respose of the proposed ethods with that of the ethod reported i [, as well as the perforace copariso of the above ethods for the regulatory cotrol proble. Obviously, the servo- Figure. Copariso for UFOPDT systes for a =a =. Solid blue lie: Method, solid black lie: Method, dotted lie: Method of [ Figure. Copariso for UFOPDT systes for a =a =.5. Solid blue lie: Method, solid black lie: Method, dotted lie: Method of [. respose obtaied by our ethod is cosiderably better. The regulatory cotrol perforace obtaied by our ethod is siilar to that obtaied by the ethod i [, i ters of settlig tie. However, our ethod produces a ore oscillatory respose with greater error. To settle this cocer, we ext apply Method with a =.5 ad b =. With this selectio, the PDF cotroller settigs obtaied by Method are P =.44, =.5, d =.4. The PDF cotroller settigs obtaied whe applyig Method with a =.5 ad b =5 are P =.4744, =.87, d =.46. Figure shows the obtaied resposes. The error i the regulatory cotrol case is ow alost equal to that obtaied by the ehaced ethod i [, although the respose is still slightly ore oscillatory. A cosiderably better respose ca be obtaied usig the iproved Method. Settig γ=5.55 ad applyig the iproved Method, with a =4.58

12 Figure. Copariso for UFOPDT systes. Solid black lie: proved Method, dotted lie: Method of [ Figure 4. Closed-loop resposes for UFOPDT odels uder ucertaity i. Other leged as i Fig.. Table 6. SE, AE ad TAE values for Exaple 6. Criterio proved Method Method of [ SE-servo.6 5. AE-servo TAE-servo.698. SE-load AE-load TAE-load ad b =.747, we obtai b =5.458, ad the cotroller settigs P =.76, =.8, d = Figure shows the obtaied resposes. Obviously, the servo-perforace obtaied by our iproved Method is the best ad produce o overshoot. The regulatory cotrol perforace is better tha that obtaied by the ethod i [. Table 6 suarizes the SE, AE ad TAE values obtaied by the two ethods i copariso. The robustess of the proposed ethods is evaluated by allowig a % ucertaity i the process gai. Figure 4 shows the resposes i this case. The ethods i copariso have siilar robust perforaces. Siilar results ca be obtaied for a % ucertaity i T or d. Fially, it is worth oticig that the ethod preseted i [ caot work for values of d/t>.5. cotrast, both Method ad (ad of course the iproved Method ) is applicable for values of d/t up to. The iterested reader ca verify that for d=.75, Method applied for a =5 ad b =, provides the PDF cotroller settigs P =.9, =7.56-5, d =.775. The PDF cotroller settigs obtaied whe applyig Method with a = 5 ad b =5 are P =.9, = , d =.775. The iproved Method, applied for a =5, b =.747, b =5.458 (γ=5.55), gives, i this case, the cotroller settigs P =.7, =7.45-4, d = All these settigs give stable closed- loop resposes, with the iproved Method ΙΙ providig the best respose. The PD cotroller settigs obtaied by the ethod i [, with the tuig paraeters ivolved havig the values 4 ad 4.4, are C =.67, τ i = ad τ D =.878. However, these settig lead to a ustable closed-loop respose. 4.7 Exaple 7 this exaple, we will exaie the efficiecy of our ethods i tuig PDF cotrollers for secod order odels with dead tie, havig a ustable pole (U SOPDT). These odels have the for G P = exp( d /[( T s )( Ts ) To this ed, cosider the process odel reported i [9, with =;T =5, T =.7, d=.99. Method is applied here by selectig a =4. ad b =. With this selectio, the PDF cotroller settigs obtaied by Method are P =5.889, =.84, d = The PDF cotroller settigs obtaied whe applyig Method with a =4. ad b =5 are P =5.96, =.56, d = The settigs of a PD cotroller tued accordig to the ethod reported i [9 are C =7.44, τ i =6.684 ad τ D =.655. Figure 5 shows the copariso of the resposes obtaied by the proposed ethods ad the ethod i [9. Obviously, the servo-perforace obtaied by our ethod is cosiderably better. However, the regulatory cotrol perforace obtaied by the ethod reported i [9 is better i ters of axiu error ad settlig tie. A better respose ca be obtaied if we apply the iproved ethod for a =7, b =., b =5. this case, the PDF cotroller settigs obtaied by the iproved Method are P =7.976, =.4, d =.67. Figure

13 Figure 5. Copariso for U SOPDT systes with a =a =4., b =, b =5. Solid blue lie: Method, solid black lie: Method, dotted lie: Method of [ Figure 6. Copariso for U SOPDT systes. Solid lie: proved Method, dotted lie: Method of [9. 6 shows a copariso of the resposes obtaied by our ethod ad that reported i [9. A cosiderably better servo respose is obtaied by our ethod. Moreover, i this case, a better regulatory cotrol perforace is also obtaied by our ethod. Fially, Figure 7 shows the regulatory cotrol perforaces obtaied by the two ethods i copariso, i the case of a % ucertaity i. Obviously, the closed-loop syste with the PDF cotroller desiged accordig to the iproved Method ca better tolerate the assued ucertaity. 4.8 Exaple 8 this exaple, we cosider a secod order plus dead tie odel with two ustable poles (U SOPDT odel) of the for G P = exp( d /[( T s )( Ts ) with =, T =, T = ad d=. [9. Method is applied here with a =6 ad b =. With this selectio, the PDF cotroller settigs obtaied by Method are P =.8, =.949, d =.875. The PDF cotroller settigs obtaied whe applyig Method with a =6 ad b =5 are P =.8, =.65, d =.9. The settigs of a PD cotroller tued accordig to the ethod reported i [9 are C =.5, τ i =.784 ad τ D = Figure 8 shows the copariso of the resposes obtaied by the proposed ethods ad the ethod i [9. Obviously, the servo-resposes obtaied by the ethods i copariso are siilar (the ethod i [9 gives a greater overshoot). However, the respose obtaied by the ethod of [9, i the case of the regulatory cotrol proble is cosiderable better tha those obtaied by the proposed ethods Figure 7. Copariso for U SOPDT systes, i ters of the regulatory cotrol perforace uder %ucertiaty i. Solid lie: proved Method, dotted lie: Method of [ Figure 8. Copariso for U SOPDT systes with a =a =6, b =, b =5. Solid blue lie: Method, solid black lie: Method, dotted lie: Method of [9.

14 To obtai a better respose, we apply the iproved ethod for a =, b =, b =5. this case, the PDF cotroller settigs obtaied by the iproved Method are P =.6, =.4, d = Figure 9 shows a copariso of the resposes obtaied by our ethod ad that reported i [9. A cosiderably better servo respose is obtaied by our ethod, while a better regulatory cotrol perforace is also achieved. Note that, i this case, the value of the SE-load criterio obtaied by the ethod of [9 is.97, while our ethod provides a respective value of.7. Fially, Figure shows the regulatory cotrol perforaces obtaied by the two ethods i copariso, i the case of a % ucertaity i. Obviously, the closed-loop syste with the PDF cotroller desiged accordig to the iproved Method ca better tolerate the assued ucertaity Figure 9. Copariso for U SOPDT systes. Solid lie: proved Method, dotted lie: Method of [ Exaple 9 this exaple, we cosider a itegratig ad ustable with dead tie (UFOPDT) process odel of the for G P = exp( d /[ s( Ts ) with =, T= ad d=. [9. Method is applied here with a =6 ad b =. With this selectio, the PDF cotroller settigs obtaied by Method are P =., =.7778, d =. The PDF cotroller settigs obtaied whe applyig Method with a =6 ad b =5 are P =.79, =.975, d =.. The settigs of a PD cotroller tued accordig to the ethod reported i [9 are C =.84, τ i =.66 ad τ D =.8. Figure shows the copariso of the resposes obtaied by the proposed ethods ad the ethod i [9. Obviously, the proposed ethods have a better perforace, particularly i the case of regulatory cotrol. 4. Exaple this exaple, we cosider a overdaped secod order plus dead tie process odel with a stable zero (overdaped SOPDTSZ), of the for GP = ( Tz s ) exp( d /[( T s )( Ts ) (6) Process odels of the for (6), which are also kow as large overshoot systes because their ope-loop dyaic behavior ca have a iitial large overshoot respose, are uch ore difficult to cotrol tha the usual first-order, secod-order or itegratig plus dead-tie systes. Very few discussios o proper cotroller tuig for such odels have bee reported i the literature, the ost recet beig that reported i [, where a foral ethod is preseted i order to obtai the settigs of PD Figure. Copariso for U SOPDT systes, i ters of the regulatory cotrol perforace uder %ucertiaty i. Solid lie: proved Method, dotted lie: Method of [ Figure. Copariso for UFOPDT systes with a =a =6, b =, b =5. Solid blue lie: Method, solid black lie: Method, dotted lie: Method of [9.

15 cotrollers of the for τ = Ds Gc c τ s τ F s the preset exaple, =, T =, T =, T z =.5, d=.5. With these syste paraeters, the settigs of a PD cotroller tued accordig to the ethod i [ are C =.485, τ i =, τ D = ad τ F =.5. Method, with a =, b =, gives the PDF cotroller paraeters as P =7.64, =6.45, d =.8. The PDF cotroller settigs obtaied by the applicatio of Method, with a =.96, b =.6, are P =6.7, = 6.68, d =.656. Figure shows the copariso of the resposes obtaied by the proposed ethods ad the ethod i [. The servo-resposes obtaied by our ethods give soe sall overshoot, but the resposes i the case of the regulatory cotrol proble are cosiderably better tha that obtaied by the ethod of [. A better Figure. Copariso for overdaped SOPDTSZ systes. Solid blue lie: Method, solid black lie: Method, dotted lie: Method of [. servo respose ca be obtaied whe applyig the iproved ethod with a =.6, b =.5, b =5 are P =8.96, =5.494, d =.776. Figure shows the copariso of the obtaied resposes. Obviously our ethod gives the better respose. 4. Exaple this exaple, we cosider a uderdaped secod order plus dead tie process odel with a stable zero (uderdaped SOPDTSZ), of the for GP = ( Tz s ) exp( d /( ω s ζωs ) with ω=, ζ=.5, Κ=, Τ z =.5, d=.5. Method is applied here with a =. ad b =.6. With this selectio, the PDF cotroller settigs obtaied by Method are P =.97, =.89, d =.5. The PDF cotroller settigs obtaied whe applyig Method with a =.69 ad b =.5 are P =.49, =.457, d =.58. The settigs of a PD cotroller tued accordig to the ethod reported i [ are C =.884, τ i =, τ D = ad τ F =.5. Figure 4 shows the copariso of the resposes obtaied by the proposed ethods ad the ethod i [. Obviously, the proposed ethods have a better perforace, particularly i the case of regulatory cotrol. A siilar perforace ca be obtaied i the case where the PDF cotroller is desiged accordig to the iproved ethod with a =.5, b =.4, b =5. this case the obtaied PDF cotroller settigs are P =.548, =.77, d = Exaple this exaple, we cosider a secod order plus dead tie process odel with two ustable poles ad a stable zero (U SOPDTSZ). The process odel cosidered has the for Figure. Copariso for overdaped SOPDTSZ systes. Solid lie: proved Method, dotted lie: Method of [ Figure 4. Copariso for uderdaped SOPDTSZ systes. Solid blue lie: Method, solid black lie: Method, dotted lie: Method of [.

16 [( T s ) ( T ) GP = ( Tz s ) exp( d / s with =, T z =5, T =, T = ad d=. [9. For this particular exaple, the ethod reported i [9 fails to provide settigs for a PD (or, i geeral, for a siple three-ter cotroller), sice, as stated i [9, the strog lead ter (5 causes tuig paraeters τ ad τ D of the PD cotroller to be egative values. For this reaso, i [9, a four-ter PD lag cotroller is ecessary to cotrol the process. However, if we apply the proposed Method with a =.5, b =.8, we obtai the PDF cotroller settigs P =.84, =.84, d =.7. Moreover, the PDF cotroller settigs obtaied whe applyig Method with a = ad b =.6 are P =.99, =.44, d =.. Figure 5 shows the closed-loop resposes obtaied whe applyig the above tree-ter cotrollers to the process odel cosidered. Obviously, the proposed ethods are capable to give acceptable three-ter cotroller settigs eve i the case where other ethods fail. 4. Exaple this exaple, we cosider a overdaped secod order plus dead tie process odel with a ustable zero (overdaped SOPDTUZ). The process odel cosidered has the for GP = ( Tz s ) exp( d /[( T s ) ( Ts ) (7) Process odels of the for (7), which are also kow as iverse respose systes because their ope-loop dyaic behavior ca have a iitial wrog way respose, are also difficult to cotrol. The iterested reader ca refer to [, [ for a review of ethods for proper cotroller tuig for such odels. particular, i [, a siple ethod is preseted i order to obtai the settigs of PD cotrollers of the for τ = Ds U C R( Y ( s ) τ s.τ Ds the preset exaple, =, T =, T =, T z =.6, d=. With these syste paraeters, the settigs of a PD cotroller tued accordig to the ethod i [ are C =.8, τ i =, τ D =. Method, with a =4., b =, gives the PDF cotroller paraeters as P =., =.497, d =.545. The PDF cotroller settigs obtaied by the applicatio of Method, with a =6, b =5, are P =.58, =.6, d =.55. Figure 6 shows the copariso of the servo resposes obtaied by the proposed ethods ad the ethod i [. The servo-resposes obtaied by our ethods give saller closed-loop jup due to the iverse respose, while Method gives a sall overshoot. Figure 7 shows the resposes obtaied i the case of regulatory cotrol. this case, all three ethods give the sae closed-loop jup while our ethods give a saller axiu error. Moreover, the SEload values i this case are obtaied as.8 for the Method of [,.965 for Method ad.776 for Method. Therefore, Method gives the better perforace. A siilar good regulatory cotrol perforace ca be obtaied by applyig the iproved Method with a =6, b = ad b =5. Cotroller paraeters i this case are obtaied as P =.74, =.78, d =.989. The SE-load value for iproved Method is.646. Figure 8 shows the obtaied resposes. 4.4 Exaple 4 this exaple, we cosider a uderdaped secod order plus dead tie process odel with a ustable zero (uderdaped SOPDTUZ). The pro Figure 5. Closed-loop resposes for U SOPDTSZ systes. Black lie: Method, blue lie: Method Figure 6. Copariso of servo-resposes obtaied for SOPDTUZ systes: Solid blue: Method, solid black: Method, dotted: Method of [.

17 Figure 7. Copariso of the regulatory cotrol resposes obtaied for SOPDTUZ systes: Solid blue lie: Method, solid black lie: Method, dotted lie: Method of [ Figure 8. Copariso of regulatory cotrol resposes obtaied for SOPDTUZ systes: Solid: proved Method, dotted: Method of [. cess odel cosidered has the for GP = ( T z s ) exp( d /( ω s ζωs ) (8) For the preset exaple =, T z =.6, d=, ω=, ζ=.5. For odels of the for (8), a ethod has bee preseted i [ for tuig a special class of PD cotrollers havig the for τ = D s U C R( Y ( s ) τ s τ s.τ D s the sequel, we will see that it is ot ecessary to resort to such a peculiar ad coplicated PD cotroller structure, as i [. particular, we will see that ore covetioal three-ter cotrollers are sufficiet to adequately cotrol a process odel of the for (8). To this ed, we ext apply Method with a =4.4, b =.8. The obtaied PDF cotroller settigs are P =.44, =., d =.487. Based o the equivalece betwee the PDF cotroller ad the ideal PD cotroller, the equivalet PD cotroller settigs are c =.44, τ =.6, τ D =.8646, while the set-poit filter trasfer fuctio () is give by G SPF (=/(.56s.6. Moreover, applicatio of the proposed Method, with a =6, b = 8.5, gives the PDF cotroller settigs P =.75, =.66, d = The settigs of the equivalet PD cotroller are c =.75, τ =., τ D =.955, while the set-poit filter has the for G SPF (=/(.54s.. Fially, the paraeters of the odified PD cotroller desiged accordig to the ethod reported i [ are c =.55, τ =.8989, τ D =.4. Figure 9 shows the obtaied resposes for both the servo ad the regulatory cotrol proble. Method gives less overshoot, while Method is coparable to the ethod of [ i ters of overshoot ad settlig tie. Our ethods give less closed-loop jup, ad i the case of regulatory cotrol, less axiu error. Moreover, the SE-load values i this case are obtaied as.67 for the Method of [,.995 for Method ad.668 for Method. Overall, our ethods give a better perforace. 4.5 Exaple 5 this exaple, we cosider a secod order plus dead-tie odel with a ustable pole ad a ustable zero (U SOPDTUZ odel). The process odel has the for GP = ( Tz s ) exp( d /[( T s ) ( Ts ) To the authors best kowledge there is o ethod i the literature for tuig three ter cotroller for such processes. Here, we test the proposed e Figure 9. Copariso for uderdaped SOPDTUZ systes: Solid blue lie: Method, solid black lie: Method, dotted lie: Method of [.